MathDB

Let

k,n

and

r

be positive integers.
(a) Let

Q(x)=x^k+a_1x^{k+1}+\cdots+a_nx^{k+n}

be a polynomial with real coefficients. Show that the function

\frac{Q(x)}{x^k}

is strictly positive for all real

x

satisfying

0<|x|<\frac1{1+\sum\limits_{i=1}^n |a_i|}

(b) Let

P(x)=b_0+b_1x+\cdots+b_rx^r

be a non zero polynomial with real coefficients. Let

m

be the smallest number such that

b_m \neq 0

. Prove that the graph of

y=P(x)

cuts the

x

-axis at the origin (i.e.,

P

changes signs at

x=0

) if and only if

m

is an odd integer.

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